Probability of Baseball Division Title

1. The problem statement, all variables and given/known data
On the morning of September 30, 1982 the won lost records of the three leading baseball teams in the Western Division of the National league were:
Braves Won: 87 Lost:72
Giants Won: 86 Lost: 73
Dodgers Won: 86 Lost: 73
Each team had 3 games remaining. All 3 of the Giants' games were with the Dodgers and the 3 remaining games of the Braves were against the Padres. Suppose that the outcomes of all remaining games are independent and each game is equally likely to be won by either participant. For each team, what is the probability that it will win the division title? If two teams tie for first place, they have a a playoff game, which each team has an equal chance of winning.

I don't even know where to start with this problem.
Since each team has an equally likely chance of winning any game, the probability that any team will win any game is 1/2.
So the probably that any team will win all three of the games they have left to play is (1/2)3. However, since the probably that they win a game is a half, the probability that they lose a game is also a half. So the probability that they win one game and lose 2 games or win 2 games and lost one game or lose all three games would still be (1/2)3.
I feel like there are a lot of cases to investigate in this problem potentially. I mean if the
Braves win all three games, they obviously will have the title no matter what the other two teams do. If the Braves win one game and either the Giants or the Dodgers win two games, there will be a tie. If the Braves win one game and the Giants or the Dodgers win 3 games, then they win, etc, etc, etc.
I don't even know if all these cases are supposed to matter in how I will get to what I want in the end. I realize that there will only be two main calculations since the Giants and the Dodgers both will have the same probability but I just don't even know where to begin on trying to find either their probability or the probability of the Braves.
Supposedly the answers are: Braves 38/64, Giants: 13/64, Dodgers: 13/64.

Any insight on to how I should go about this would be greatly appreciated!
Thanks!

How about laying out the sample space for possible ways of having a team win, e.g., one way the Braves would clinch if they won 4 games. What if there is a tie; what would then have to happen for someone to win?